It depends on how you define the integers. After creating the objects, defining which objects are unequal and equal is exactly what gives them the useful structure and behavior you want. That is, it's not the equality axioms but the axioms of the integers that say the interesting things about what's equal and what isn't. See this link
https://en.m.wikipedia.org/wiki/Peano_axiomsfor one example of axioms that give the formalization you are looking for (note these build just 0,1,2,3,..., but more objects and structure can then be added to create "all the integers" along with a sensible notion of equality among them)
